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1. 𝐿₁-distortion of Wasserstein metrics: A tale of two dimensions

   https://doi.org/10.1090/btran/143  

   Baudier, F.; Gartland, C.; Schlumprecht, Th. (August 2023, Transactions of the American Mathematical Society, Series B)

   By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid $\{0,1,\dots <\#comment/>,n{\}}^2$has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log <\#comment/>n}$. We provide a new “dimensionality” interpretation of Kislyakov’s argument, showing that if $\{G_n{\}}_{n=1}^{\mathrm{\infty <\#comment/>}}$is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number $\mathrm{\delta <\#comment/>}\in <\#comment/>[2,\mathrm{\infty <\#comment/>})$, then the 1-Wasserstein metric over $G_n$has $L_1$-distortion bounded below by a constant multiple of $(\log <\#comment/>|G_n|{)}^{\frac{1}{\mathrm{\delta <\#comment/>}}}$. We proceed to compute these dimensions for $\oslash <\#comment/>$-powers of certain graphs. In particular, we get that the sequence of diamond graphs $\{{\mathsf{D}}_n{\}}_{n=1}^{\mathrm{\infty <\#comment/>}}$has isoperimetric dimension and Lipschitz-spectral dimension equal to 2, obtaining as a corollary that the 1-Wasserstein metric over ${\mathsf{D}}_n$has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log <\#comment/>|{\mathsf{D}}_n|}$. This answers a question of Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence of $L_1$-embeddable graphs whose sequence of 1-Wasserstein metrics is not $L_1$-embeddable. 

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2. Braid group action and quasi-split affine 𝚤quantum groups I

   https://doi.org/10.1090/ert/657  

   Lu, Ming; Wang, Weiqiang; Zhang, Weinan (October 2023, Representation Theory of the American Mathematical Society)

   This is the first of our papers on quasi-split affine quantum symmetric pairs $(\stackrel{~<\#comment/>}{\mathbf{U}}\left(\stackrel{^<\#comment/>}{\mathfrak{g}}\right),{\stackrel{~<\#comment/>}{\mathbf{U}}}^{\mathrm{ı<\#comment/>}})$, focusing on the real rank one case, i.e., $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_3$equipped with a diagram involution. We construct explicitly a relative braid group action of type $A_2^{\left(2\right)}$on the affine $\mathrm{ı<\#comment/>}$quantum group ${\stackrel{~<\#comment/>}{\mathbf{U}}}^{\mathrm{ı<\#comment/>}}$. Real and imaginary root vectors for ${\stackrel{~<\#comment/>}{\mathbf{U}}}^{\mathrm{ı<\#comment/>}}$are constructed, and a Drinfeld type presentation of ${\stackrel{~<\#comment/>}{\mathbf{U}}}^{\mathrm{ı<\#comment/>}}$is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine $\mathrm{ı<\#comment/>}$quantum groups in the sequels. 

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3. Algebraic cobordism and a Conner–Floyd isomorphism for algebraic K-theory

   https://doi.org/10.1090/jams/1045  

   Annala, Toni; Hoyois, Marc; Iwasa, Ryomei (March 2024, Journal of the American Mathematical Society)

   We formulate and prove a Conner–Floyd isomorphism for the algebraic K-theory of arbitrary qcqs derived schemes. To that end, we study a stable $\mathrm{\infty <\#comment/>}$-category of non- ${\mathbb{A}}^1$-invariant motivic spectra, which turns out to be equivalent to the $\mathrm{\infty <\#comment/>}$-category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We prove that this $\mathrm{\infty <\#comment/>}$-category satisfies ${\mathbb{P}}^1$-homotopy invariance and weighted ${\mathbb{A}}^1$-homotopy invariance, which we use in place of ${\mathbb{A}}^1$-homotopy invariance to obtain analogues of several key results from ${\mathbb{A}}^1$-homotopy theory. These allow us in particular to define a universal oriented motivic ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-ring spectrum $\mathrm{M}\mathrm{G}\mathrm{L}$. We then prove that the algebraic K-theory of a qcqs derived scheme $X$can be recovered from its $\mathrm{M}\mathrm{G}\mathrm{L}$-cohomology via a Conner–Floyd isomorphism\[ ${\mathrm{M}\mathrm{G}\mathrm{L}}^{\ast <\#comment/>\ast <\#comment/>}\left(X\right){\otimes <\#comment/>}_{\mathrm{L}}\mathbb{Z}\left[{\mathrm{\beta <\#comment/>}}^{\pm <\#comment/>1}\right]\simeq <\#comment/>\mathrm{K}{}^{\ast <\#comment/>\ast <\#comment/>}\left(X\right),$\]where $\mathrm{L}$is the Lazard ring and $\mathrm{K}{}^{p,q}\left(X\right)=\mathrm{K}{}_{2q-<\#comment/>p}\left(X\right)$. Finally, we prove a Snaith theorem for the periodized version of $\mathrm{M}\mathrm{G}\mathrm{L}$. 

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4. The 𝑚=2 amplituhedron and the hypersimplex: Signs, clusters, tilings, Eulerian numbers

   https://doi.org/10.1090/cams/23  

   Parisi, Matteo; Sherman-Bennett, Melissa; Williams, Lauren (July 2023, Communications of the American Mathematical Society)

   The hypersimplex ${\mathrm{\Delta <\#comment/>}}_{k+1,n}$is the image of the positive Grassmannian $G{r}_{k+1,n}^{\ge <\#comment/>0}$under the moment map. It is a polytope of dimension $n-<\#comment/>1$in ${\mathbb{R}}^n$. Meanwhile, the amplituhedron ${\mathcal{A}}_{n,k,2}\left(Z\right)$is the projection of the positive Grassmannian $G{r}_{k,n}^{\ge <\#comment/>0}$into the Grassmannian $G{r}_{k,k+2}$under a map $\stackrel{~<\#comment/>}{Z}$induced by a positive matrix $Z\in <\#comment/>Ma{t}_{n,k+2}^{>0}$. Introduced in the context ofscattering amplitudes, it is not a polytope, and has full dimension $2k$inside $G{r}_{k,k+2}$. Nevertheless, there seem to be remarkable connections between these two objects viaT-duality, as conjectured by Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]. In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting outpositroid polytopes—images of positroid cells of $G{r}_{k+1,n}^{\ge <\#comment/>0}$under the moment map—translate into sign conditions characterizing the T-dualGrasstopes—images of positroid cells of $G{r}_{k,n}^{\ge <\#comment/>0}$under $\stackrel{~<\#comment/>}{Z}$. Moreover, we subdivide the amplituhedron intochambers, just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We use these properties to prove the main conjecture of Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]: a collection of positroid polytopes is a tiling of the hypersimplex if and only if the collection of T-dual Grasstopes is a tiling of the amplituhedron ${\mathcal{A}}_{n,k,2}\left(Z\right)$for all $Z$. Moreover, we prove Arkani-Hamed–Thomas–Trnka’s conjectural sign-flip characterization of ${\mathcal{A}}_{n,k,2}$, and Łukowski–Parisi–Spradlin–Volovich’s conjectures on $m=2$cluster adjacencyand onpositroid tilesfor ${\mathcal{A}}_{n,k,2}$(images of $2k$-dimensional positroid cells which map injectively into ${\mathcal{A}}_{n,k,2}$). Finally, we introduce new cluster structures in the amplituhedron. 

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5. On the Size of the Singular Set of Minimizing Harmonic Maps

   https://doi.org/10.1090/memo/1519  

   Mazowiecka, Katarzyna; Miśkiewicz, Michał; Schikorra, Armin (October 2024, Memoirs of the American Mathematical Society)

   We consider minimizing harmonic maps $u$from $\mathrm{\Omega <\#comment/>}\subset <\#comment/>{\mathbb{R}}^n$into a closed Riemannian manifold $\mathcal{N}$and prove: 1. an extension to $n\ge <\#comment/>4$of Almgren and Lieb’s linear law. That is, if the fundamental group of the target manifold $\mathcal{N}$is finite, we have\[ ${\mathcal{H}}^{n-<\#comment/>3}(\mathrm{sing}<\#comment/>u)\le <\#comment/>C{\int <\#comment/>}_{\mathrm{\partial <\#comment/>}\mathrm{\Omega <\#comment/>}}|{\mathrm{\nabla <\#comment/>}}_{T}u{|}^{n-<\#comment/>1}\mathrm{d}{\mathcal{H}}^{n-<\#comment/>1};$\]2. an extension of Hardt and Lin’s stability theorem. Namely, assuming that the target manifold is $\mathcal{N}={\mathbb{S}}^2$we obtain that the singular set of $u$is stable under small $W^{1,n-<\#comment/>1}$-perturbations of the boundary data. In dimension $n=3$both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$with $s\in <\#comment/>(\frac{1}{2},1]$and $p\in <\#comment/>[2,\mathrm{\infty <\#comment/>})$satisfying $sp\ge <\#comment/>2$. We also discuss sharpness. 

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